code: def _nls_err_func(
      tensor_elements,
      design_matrix,
      data,
      Diso=3e-3,
      weighting=None,
      sigma=None,
      cholesky=False,
      f_transform=False,
  ):
      """Error function for the non-linear least-squares fit of the tensor water
      elimination model.
  
      Parameters
      ----------
      tensor_elements : array (8, )
          The six independent elements of the diffusion tensor followed by
          -log(S0) and the volume fraction f of the water elimination
          compartment. Note that if cholesky is set to true, tensor elements are
          assumed to be written as Cholesky's decomposition elements. If
          f_transform is true, volume fraction f has to be converted to
          ft = arcsin(2*f - 1) + pi/2
      design_matrix : array
          The design matrix
      data : array
          The voxel signal in all gradient directions
      Diso : float, optional
          Value of the free water isotropic diffusion. Default is set to 3e-3
          $mm^{2}.s^{-1}$. Please adjust this value if you are assuming different
          units of diffusion.
      weighting : str (optional).
           Whether to use the Geman-McClure weighting criterion (see [1]_
           for details)
      sigma : float or float array (optional)
          If 'sigma' weighting is used, we will weight the error function
          according to the background noise estimated either in aggregate over
          all directions (when a float is provided), or to an estimate of the
          noise in each diffusion-weighting direction (if an array is
          provided). If 'gmm', the Geman-Mclure M-estimator is used for
          weighting.
      cholesky : bool, optional
          If true, the diffusion tensor elements were decomposed using Cholesky
          decomposition. See fwdti.nls_fit_tensor
          Default: False
      f_transform : bool, optional
          If true, the water volume fraction was converted to
          ft = arcsin(2*f - 1) + pi/2, insuring f estimates between 0 and 1.
          See fwdti.nls_fit_tensor
          Default: True
      """
      tensor = np.copy(tensor_elements)
      if cholesky:
          tensor[:6] = cholesky_to_lower_triangular(tensor[:6])
  
      if f_transform:
          f = 0.5 * (1 + np.sin(tensor[7] - np.pi / 2))
      else:
          f = tensor[7]
  
      # This is the predicted signal given the params:
      y = (1 - f) * np.exp(
          np.dot(design_matrix, tensor[:7])
      ) + f * np.exp(
          np.dot(
              design_matrix,
              np.array([Diso, 0, Diso, 0, 0, Diso, tensor[6]]),
          )
      )
  
      # Compute the residuals
      residuals = data - y
  
      # If we don't want to weight the residuals, we are basically done:
      if weighting is None:
          # And we return the SSE:
          return residuals
      se = residuals**2
      # If the user provided a sigma (e.g 1.5267 * std(background_noise), as
      # suggested by Chang et al.) we will use it:
      if weighting == "sigma":
          if sigma is None:
              e_s = "Must provide sigma value as input to use this weighting"
              e_s += " method"
              raise ValueError(e_s)
          w = 1 / (sigma**2)
  
      elif weighting == "gmm":
          # We use the Geman-McClure M-estimator to compute the weights on the
          # residuals:
          C = 1.4826 * np.median(
              np.abs(residuals - np.median(residuals))
          )
          with warnings.catch_warnings():
              warnings.simplefilter("ignore")
              w = 1 / (se + C**2)
              # The weights are normalized to the mean weight (see p. 1089):
              w = w / np.mean(w)
  
      # Return the weighted residuals:
      with warnings.catch_warnings():
          warnings.simplefilter("ignore")
          return np.sqrt(w * se)
  
imports: import warnings
  import numpy
  import scipy.optimize
  from dipy.reconst.base import ReconstModel
  from dipy.reconst.dti import TensorFit, design_matrix, decompose_tensor, _decompose_tensor_nan, from_lower_triangular, lower_triangular
  from dipy.reconst.dki import _positive_evals
  from dipy.reconst.vec_val_sum import vec_val_vect
  from dipy.core.ndindex import ndindex
  from dipy.core.gradients import check_multi_b
  from dipy.reconst.multi_voxel import multi_voxel_fit
label_desc: Use image node_id 4 for calling a global function with example usage: _nls_err_func(tensor_elements, design_matrix, data, Diso, weighting, sigma, cholesky, f_transform) and returns: residuals, np